Weighted Parameterized Correlation Coefficients of Indeterminacy Fuzzy Multisets and Their Multicriteria Group Decision Making Method with Different Decision Risks

Cheng Du and Jun Ye

1Beijing Urban Construction Group Co.,Ltd.,Beijing,100088,China

2School of Civil and Environmental Engineering,Ningbo University,Ningbo,315211,China

ABSTRACT Real-life data introduce noise,uncertainty,and imprecision to statistical projects;it is advantageous to consider strategies to overcome these information expressions and processing problems.Neutrosophic(indeterminate)numbers can flexibly and conveniently represent the hybrid information of the partial determinacy and partial indeterminacy in an indeterminate setting,while a fuzzy multiset is a vital mathematical tool in the expression and processing of multi-valued fuzzy information with different and/or same fuzzy values.If neutrosophic numbers are introduced into fuzzy sequences in a fuzzy multiset,the introduced neutrosophic number sequences can be constructed as the neutrosophic number multiset or indeterminate fuzzy multiset.Motivated based on the idea,this study first proposes an indeterminate fuzzy multiset,where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values.Then,we propose the parameterized correlation coefficients of indeterminate fuzzy multisets based on the de-neutrosophication of transforming indeterminate fuzzy multisets into the parameterized fuzzy multisets by a parameter(the parameterized de-neutrosophication method).Since indeterminate decision-making issues need to be handled by an indeterminate decision-making method,a group decision-making method using the weighted parameterized correlation coefficients of indeterminate fuzzy multisets is developed along with decision makers’different decision risks(small,moderate,and large risks)so as to handle multicriteria group decision-making problems in indeterminate fuzzy multiset setting.Finally,the developed group decision-making approach is used in an example on a selection problem of slope design schemes for an open-pit mine to demonstrate its usability and flexibility in the indeterminate group decision-making problem with indeterminate fuzzy multisets.

KEYWORDS Indeterminate fuzzy multiset;parameterized correlation coefficient;multicriteria group decision making;neutrosophic number;slope design scheme

1 Introduction

Fuzzy sets(FSs)and fuzzy multisets(FMs)are vital mathematical tools in the expression and processing of fuzzy information since there are uncertainty and vagueness in many real-life problems.In the fuzzy theory,the FS presented by Zadeh[1]is depicted by a degree of membership,which usually contains almost one occurrence of each element in FS.Since then,fuzzy sets have wildly applied in various areas[2–9].As an extension of FS,Yager[10]and Miyamoto[11]proposed the concept of fuzzy bag/FM,where each element in a universe set can be repeated more than once with the different and/or identical membership values.Then,FMs have been used for various applications in many areas[12–15].Recently,El-Azab et al.[16]proposed a correlation coefficient of FMs as the extension of a correlation coefficient of FSs[17]and gave its application in the setting of FMs.Further,Das[18]put forward weighted fuzzy soft multiset(WFSM)and introduced an adjustable approach to the WFSMs-based decision-making method in uncertain environment.Furthermore,an interval-valued fuzzy multiset(IvFM)[19]was presented as the extension of FMs,where each element in a universe set can be repeated more than once with the different and/or identical membership values.In the hesitant fuzzy environment,then,a hesitant fuzzy set(HFS)and an interval-valued hesitant fuzzy set(IvHFS)[20]were represented by a group of different fuzzy values,but they cannot reflect the same fuzzy values in a group of hesitant fuzzy values.Under indeterminate and inconsistent environments,the neutrosophic multisets with the same or different neutrosophic components were proposed based on truth,falsity and indeterminacy membership functions and used for the applications in physical processes[21,22].

Due to the vagueness and indeterminacy of human cognition/judgments in complicated real problems,the human judgment may contain the hybrid information of the partial determinacy and partial indeterminacy.However,FMs and fuzzy soft multisets cannot represent the indeterminate fuzzy sequences,while a neutrosophic/indeterminate number(NN/IN)[23–25]is composed of its determinate termdand its indeterminate termuIwith indeterminacyI∈[infI,supI]and denoted asz=d+uIford,u∈?.In indeterminate problems,since NN can be considered as a changeable interval number/value regarding a changeable range/value ofI∈[infI,supI],it can flexibly and conveniently describe the hybrid information of the partial determinacy and partial indeterminacy,and also easily indicate a family of interval numbers(zi=[di+uiinfI,di+uisupI]fori= 1,2,...,q)depending on a group of specified indeterminate ranges ofI∈[infI,supI],which shows its highlighting advantage in the indeterminate information expression.Therefore,NNs have been wildly used in many areas,such as rock mechanics[26,27],decision making[28],fault diagnosis[29],and linear and nonlinear programming[30,31].

Since FMs,HFSs,IvFMs,and IvHFSs cannot express indeterminate fuzzy sequences by the indeterminate membership values in group decision making(GDM)problems with indeterminate information,various measure algorithms of FMs,HFSs,IvFMs,and IvHFSs cannot also deal with such GDM problems with the indeterminate fuzzy sequences and indeterminate decision risks of decision makers because indeterminate decision-making problems need to be solved by an indeterminate decision making method.If NNs/INs are introduced into fuzzy sequences in FMs and IvFMs,we can produce indeterminate/NN fuzzy sequences from the different and/or identical indeterminate membership values with some indeterminate ranges and construct an indeterminate fuzzy multiset(IFM)so as to present the parameterized correlation coefficients(PCCs)of IFMs by a parameterized de-neutrosophication method of IFM for handling indeterminate GDM problems with decision makers’different decision risks.Motivated by these new ideas,this study proposes IFM,where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values,and then introduces two PCCs of IFMs based on the de-neutrosophication of transforming IFMs into the parameterized fuzzy multisets(PFMs)by a parameterλ∈[0,1](i.e.,the parameterized de-neutrosophication method)and their GDM approach with decision makers’different decision risks(small,moderate and large risks forλ= 0,0.5,1)to solve indeterminate GDM problems in IFM setting.

However,main contributions of this study are summarized as follows:

(1)The proposed IFM contains various families of FMs,HFSs,IvFMs,and IvHFSs depending on the indeterminate ranges and values ofIand provides an expression form of the different and/or identical indeterminate membership values that HFS and IvHFS cannot express.

(2)The proposed two PCCs of IFMs based on the parameterized de-neutrosophication method can provide effective mathematical models for solving indeterminate decisionmaking problems because the correlation coefficient of FMs cannot deal with indeterminate decision-making issues.

(3)The developed GDM method based on the two PCCs can solve indeterminate GDM problems,such as the selecting problem of slope design schemes(SDSs)for an open pit mine with decision makers’different decision risks in the actual indeterminate application and show its highlighting advantages of the practicability and flexibility in IFM setting.

This paper is composed of the following parts.Section 2 introduces preliminaries of FMs.Section 3 proposes IFM and two PCCs of IFMs based on the de-neutrosophication of transforming IFMs into PFMs by a parameterλ∈[0,1].Section 4 develops a GDM method using PCCs of IFMs along with decision makers’different decision risks(small,moderate and large risks forλ=0,0.5,1)in IFM setting.In Section 5,the developed GDM method is used in a GDM example on a selection problem of SDSs for an open pit mine with decision makers’different decision risks to demonstrate the usability and flexibility of the developed GDM method in IFM setting.Conclusions and further research are contained in Section 6.

2 Preliminaries of FMs

3 IFMs and Their PCCs

4 GDM Approach Based on the Weighted PCCs in IFM Setting

5 GDM Example

Regarding an open pit mine,the design problem of the slopes is one of the major challenges at mine planning and operation[32].It requires specialized knowledge of the geology,which is often complex,vague,and indeterminate/variable in the structural and material properties of orebodies.Regarding this indeterminate issue,we provide a GDM example on the selection problem of SDSs for an open pit mine to show the usability and flexibility of the developed GDM method with different decision risks of decision makers in IFM setting.

A mining company wants to select the best SDS/alternative regarding an open pit mine.Assume a set of four potential SDSs(alternatives)G= {g1,g2,g3,g4} for the open pit mine is specified preliminarily by experts/designers.Then,the four alternatives must be satisfactorily assessed over the three criteria/indices:economy(cost and construction period)(c1),technology(reliability,effectiveness,construction difficulty,maintenance difficulty,and safety)(c2),and environment(c3).Considering the importance of the three criteria,the experts/designers give the weight vector of the three criteria byη=(0.3,0.4,0.3).

Table 1:Decision results based on with decision makers’three decision risks for λ=0,0.5,1

Table 1:Decision results based on with decision makers’three decision risks for λ=0,0.5,1

λ Nλ1(Zi,Z?)Ranking orderThe best SDS λ= 00.9951,0.9953,0.9963,0.9960g3 > g4 > g2> g1g3 λ= 0.50.9954,0.9970,0.9978,0.9979g4 > g3 > g2> g1g4 λ= 10.9950,0.9966,0.9983,0.9986g4 > g3 > g2> g1g4

Table 2:Decision results based on with decision makers’three decision risks for λ=0,0.5,1

Table 2:Decision results based on with decision makers’three decision risks for λ=0,0.5,1

λ Nλ2(Zi,Z?)Ranking orderThe best SDS λ= 00.6633,0.6667,0.7267,0.7500g4 > g3 > g2> g1g4 λ= 0.50.7517,0.7483,0.7883,0.8233g4 > g3 > g1> g2g4 λ= 10.8400,0.8300,0.8500,0.8967g4 > g3 > g1> g2g4

Table 3:Characteristic comparison between IFM and the classical FM[16]

Especially,the existing correlation coefficient of FMs[16]is only a special case of the proposed PCC of IFMs whenλonly takes a specified value.Obviously,the decision-making approach based on the correlation coefficient of FMs[16]is also a special case of the developed GDM method in IFM setting.In their decision making process,the existing decision making approach cannot contain decision makers’different decision risks sine FMs cannot contain NNs(changeable/indeterminate interval values),then it lacks the decision making flexibility in the indeterminate scenarios,while the developed GDM method can contain decision makers’different decision risks for satisfying some suitable indeterminacy by using the decision information of PFMs in the setting of IFMs and also indicate the flexible decision making advantage with various indeterminate decision risks in the indeterminate problem.Therefore,the proposed GDM method can overcome difficult/tricky decision-making problems of the existing FM decision making method in indeterminate issues because indeterminate decision-making issues need to be solved by utilizing an indeterminate decision-making method generally.However,this study proposes the GDM method by using the weighted PCCs of IFMs to solve such an indeterminate multicriteria GDM problem with decision makers’different decision risks(the small,moderate,and large risks forλ= 0,0.5,1)for the first time in IFM setting.

However,the information expression,correlation coefficient and GDM methods of IFMs proposed in this study are superior to existing methods[16].

6 Conclusion

Regarding the indeterminate multi-values of fuzzy arguments in indeterminate GDM setting,the proposed IFM in this paper presented a more flexible and useful information expression form as it addresses the indeterminacy of multi-fuzzy values corresponding to various indeterminate ranges/values of the indeterminacyI∈[infI,supI].Then,the presented PCCs of IFMs based on the de-neutrosophication of transforming IFMs into PFMs by a parameterλ∈[0,1]provided effective mathematical models for the flexible GDM method in indeterminate environment.The developed multicriteria GDM method using the weighted PCCs of IFMs along with decision makers’different decision risks(the small,moderate,and large risks forλ= 0,0.5,1)provided an effect and reasonable decision way for handling indeterminate multicriteria GDM problems in IFM setting.A GDM example of selecting the best SDS for an open pit mine was given in the environment of IFMs to demonstrate the suitability and flexibility of the developed GDM method.Then,the decision results indicated the influence of decision makers’different decision risks on the ranking order of alternatives and the best one.Therefore,the developed multicriteria GDM method can reflect its outstanding advantages of decision versatility and flexibility when dealing with indeterminate GDM problems under various decision risks.

In future research,this original work will be extended to GDM problems in construction design schemes,project management,and slope stability evaluation,and analyses under IFM environment.

Funding Statement:The authors received no specific funding for this study.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.